Iphone 5 doesn't want to turn on?

Last week I restored my iphone 5 and I also updated it, it worked normally but the only problem was that battery was dying pretty fast. After charging it I turned it off and after a day when I tried to turn it on again it just didn't want to do that. I thought that battery died, I tried to charge it but there was no signs of charging, I tried turning it on, even pressed home button and power button but nothing was appearing, it was completely dead. Earlier I took it to this store and tried to change the charger and also to check if it was the battery but even after that phone stayed dead. I don't know what could be the problem. I tried everything. I can't turn it on, power-home button is not working obviously, when I try to connect it to itunes it doesn't recognize it at all (like it is not even connected), I tried changing usb and battery and still nothing. Any ideas?

There's definitely a problem with the battery. There seems to be some defect that makes it faulty. If I were you, I would take it straight to the nearest repair store and explain to them the problem. I'm sure they'll understand, plus they won't charge you extra compared to the Genius Bar at an Apple store that takes time and usually the workers there can't be bothered to diagnose the problem (unless money is involved + insurance).